Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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The tangent space of a vector space

I'm trying to show that there is a canonical isomorphism between a finite-dimensional vector space $V$ (regarded as a $C^\infty$ manifold) and its tangent space $T_vV, v\in V$, without using a basis, ...
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34 views

Is there a Mobius (infinite) cylinder?

In order to understand the question of the title I need to understand another thing first. If we consider the Mobius band, locally, for a $U_i \subset S^1$, where $S^1$ is the base space, the bundle ...
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Whether there is easy way to compute $R_{ij}=\frac{1}{2}Rg_{ij}$ in 2-dimension

In 2-dimensional Riemann manifold ,Ricci curvature is given by $$ R_{ij}=\frac{1}{2}Rg_{ij} $$ My PDE teachers teach me to compute it by the way. $$ R_{11}=g^{ij}R_{1i1j}=g^{22}R_{1212} \\ ...
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References for metrics questions in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
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172 views
+50

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
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22 views

$1$-form not contingent upon selection of local frame

Say we have $V$ a rank $v$ $\mathbb{R}$-vector bundle over $X$, which is a $k$-dimensional manifold, with connection $\nabla$. Denote $\pi: L \to X$ be the associated linear frame bundle. This is a ...
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11 views

Covering number of the set of $n_1\times n_2$ matrices of rank at most $r$

What is the covering number of the set of $n_1\times n_2$ matrices of rank at most $r$? We know that the dimension of the set is $r(n_1+n_2-r)$. Thus, the covering number $N(\rho)\le C ...
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Metric compatibility of induced connection on submanifold of $\mathbb{R}^{n+1}$

Let $M\subset \mathbb{R}^{n+1}$ be a smooth submanifold with $\dim M =n $. Let $g$ be the induced metric on $M$ from the Standard metric on $\mathbb{R}^{n+1}$. Now, we define a connection on $TM$ by ...
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31 views

Show that the $n\times n$ matrices with determinant $=1$ forms a $C^1$ surface of dimension $n^2-1\in \mathbb{R^{n^2}}$

I am told that I need to find a path $c(t)$ such that $c(t)=x(t), X(0)=x \forall X s.t. det X=1$. So I can show that $d/dt(f(c(t))$ at $t=0=[d_{f(c(t))}f](c'(t))]\ne 0$ My problem is how to ...
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Riemann Curvature tensor for surfaces

Let $M$ be a regular surface on $\mathbb{R}^3$. I am trying to express the Riemann's curvature tensor: $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ respect $R(\vec x_i,\vec x_j)\vec ...
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Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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25 views

Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
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25 views

Fundamental Group and DeRahm Cohomology from Group of Covering Transformations

Old qual problem here, test tomorrow in topology and we barely got to DeRahm Cohomology so I'm not sure how to do this. Let $G$ be the group of transformations of $\mathbb{R}^3$ generated by ...
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23 views

Question Regarding Proof of Taylor Remainder Theorem in Tu's “An Introduction to Manifolds”

The statement: Let $f$ be a $C^{\infty}$ function on an open set $U\subseteq \mathbb{R}^n$ which is star shaped with respect to a point $p=(p^1,...,p^n) \in U$. Then there are functions ...
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14 views

Show that $x_1^2+x_2^2+…+x_n^2$ defines a $C^1 (n-1 dim)$ surface in $\mathbb{R^n}$. Compute tangent space at every point

I am not sure what the idea is behind this. There is a theorem that states if a map $F:\mathbb{R}^n\to\mathbb{R}^{n-m}$ such that $dF(x)$ has rank $n-m$ at every point on a level set then that level ...
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1answer
29 views

Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
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1answer
19 views

Sufmanifold with prescribed first and second fundamental form

Is there a $2$ - dimensional submanifold $S$ of $\mathbb{R}^3$ which can be parametrized with $x : U \subset \mathbb{R}^2 \to S$ such that : $E=G=1$, $F=0$, $e=-g=1$ and $f=0$ Where $E,F,G$ and ...
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30 views

Any oriented surface is orientable - why can we select such an atlas?

Definitions and notations: Given a surface $S$ and a surface patch $\sigma: U \subset \Bbb R^2 \to \Bbb R^3$ of $S$, we define the standard unit normal of $\sigma$ at $p$ to be (where ...
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1answer
16 views

Show every Mobius transformation $T(z)=\frac{\alpha z+ \beta}{\bar \beta z+ \bar \alpha}$ acts as an isometry of the hyperbolic disk

Consider the unit disk $\mathbb{D}=\{z: |Z| < 1\} \subset \mathbb{C}$ equipped with the hyperbolic metric $g$ induced by $1$ form $ds=\frac{|dz|}{(1-|z|^2)}$ I am trying to show that every Mobius ...
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122 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
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109 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
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351 views

Circular Helicoid

A helicoid has the following parametric equation: $$ r(u,v)=\left \langle v\cos u,v\sin u,cu \right \rangle,\qquad u,v,c\in\mathbb{R}. $$ In ruled form, $$r(u,v)=\alpha(u)+v\Lambda(u),$$ it has ...
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151 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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*understanding* covariance vs. contravariance & raising / lowering

There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
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Can you determine the length of a curve by the lengths of its projections onto planes?

If $\Gamma \subset \mathbb R^n$ is 1-rectifiable, then its Hausdorff measure is equal to its integralgeometric measure. That is, $$\mathcal H^1(\Gamma) = \int\limits_{G(1,\mathbb R^n)} \int\limits_K ...
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Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
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19 views

Prove that this is a smooth surface

S is the surface satisfying $$f(x, y, z) = z^2 + (\sqrt{x^2+y^2}-a)^2 -r^2 =0$$ where $a,r\in\mathbb{R}$ Prove that $S$ is a smooth surface. Do we differentiate with respect to $x, y$ and $z$ ...
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170 views

Gaussian curvature and mean curvature sufficient to characterize a surface?

Is the knowledge of Gaussian and mean curvature (and thus of the principal curvatures) sufficient to characterize a surface uniquely? If not, is there another geometric quantity one can add to obtain ...
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2answers
404 views

Gentle introduction to quasi-geodesics

Compared to the concept of geodesics the concept of quasi-geodesics seems to be substantially harder to grasp and digest. I was given a promising hint to the concept of quasi-geodesics here but the ...
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1answer
2k views

Riemann, Ricci curvature tensor and Ricci scalar of the n dimensional sphere

I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the n sphere $x_0^2 + x_1^2 + ....+x_n^2=R^2$, whose metric is $$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
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1answer
162 views

Zeros of the second fundamental form

Let $ f:M \rightarrow N $ be a minimal immersion (of arbitrary codimension or an hypersurface if it is necessary) and let $ |A| $ be the norm of its second fundametal form.If $ A $ is not identically ...
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36 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
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31 views

Ricci flow on compact surfaces flows the metric conformally

The (normalized) Ricci flow on compact surfaces is given by $$\frac{\partial}{\partial t}g_{ij}=(r-R)g_{ij}\text{ ,}$$ and in the beginning of Hamilton's paper on the topic he points out that since ...
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References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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1answer
382 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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1answer
15 views

Why the well-defined of Gauss map depends on surface is orientable?

Let $S$ is a surface. Define a mapping $g:S\rightarrow S^2\subset R^3$ of $S$ into the unit sphere $S^2$ , associating to every $p\in S$ a unit vector $N(p)\in S^2$ normal to $T_pS$. Why the ...
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39 views

Reference Request: Differential Geometry Book [on hold]

What is a good self study book in Differential Geometry. Keep in mind I won't have the advantage of being able to ask a professor any questions.
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1answer
26 views

assumptions for existence of envelope of a family of curves

Given a family of curves $F(x, y, c) = 0$ for $c$ in a range of real numbers, the envelope $E$ is the curve tangent to every member of the family. I see that it is defined by the solution of $F(x, ...
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1answer
30 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
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1answer
102 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
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How to reparametrize a geodesic such that $\nabla_V V = 0$

In Nakahara's section on geodesics he says that $\nabla_V V = fV$ can be reparametrized to give $\nabla_V V = 0$ if we use the reparametrize the tangent vector components $t \rightarrow t'$ such ...
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1answer
152 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
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1answer
224 views

Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
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1answer
287 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
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+100

Show that , for all $(s_{0},t_{0})\in [0,1]\times [0,a]$, the curves $s\to f(s,t_{0})$, $t\to f(s_{0},t)$ are orthogonals.

Let $f:[0,1]\times [0,a]\to M$ a parameterized surface such that for all $t_{0}\in[0,a]$, the curve $s\to f(s,t_{0})$, $s\in [0,1]$, is a parameterized geodesic by arc lenght , orthogonal to the ...
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1answer
14 views

Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
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3answers
196 views

What is the relation between dx in elementary calculus and dx in differential geometry?

I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is ...
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3answers
583 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be ...
5
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1answer
203 views

characterizing semi-Riemannian spaces of constant curvature

How does one characterize $n$-dimensional semi-Riemannian spaces of constant curvature? By "characterize," I mean giving both a definition and some insight into how the possibilities work out in ...
4
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1answer
129 views

Symplectic Chart

I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following: If $B$ is a zero neighborhood in Banach space. ...